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I was wondering why we say that a probability measure $P$ is defined on $(\Omega, \mathcal{F})$, being $\Omega$ the sample space and $\mathcal{F}$ one sigma-algebra, when actually we have that $P$ is a mapping $P:\mathcal{F}\rightarrow [0,1]$. I am not very familiar with the measurement theory and maybe that's why I am asking this (it could be a very trivial question)... But given that actually $\Omega\in\mathcal{F}$ why not to say that $P$ is defined on $\mathcal{F}$, I don't see the point of including $\Omega$ explicitly. Thanks in advance, and forgive my ignorance on the matter.

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    $\begingroup$ You are correct in that $P$ is defined on the measurable subsets of $\Omega$, that is, the elements of $\mathcal F$, so $(\Omega,\mathcal F)$ is slightly redundant, but this is the traditional wording to insist upon the dependence on the $\sigma$-algebra. $\endgroup$ – Xi'an Mar 27 at 9:02

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